Order of vertices added using Python script and bmesh
Short answer is no, however...

BMesh only ensure vertices are added in-order when they are added to a newly created (or converted) mesh (which is OK for importers and geometry generators).
Once vertices are removed you cant rely on this.

Best use a list of your own to keep a reference to vertices added if you need this.

Sorting

if your really need to ensure some order, you can sort verts using the same method used with Python lists eg:

bm.verts.sort(key=lambda v: v.co.x)


However it would be inefficient to sort vertices every time you add once.